Problem Solving - (Mathematical Processes Standard)

Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and communicate mathematical ideas.

By the end of the K - 2 program:

Use a variety of strategies to understand problem situations; e.g., discussing with peers, stating problems in own words, modeling problems with diagrams or physical materials, identifying a pattern
Identify and restate in own words the question or problem and the information needed to solve the problem.
Generate alternative strategies to solve problems.
Evaluate the reasonableness of predictions, estimations and solutions.
Explain to others how a problem was solved.
Draw pictures and use physical models to represent problem situations and solutions
Use invented and conventional symbols and common language to describe a problem situation and solution
Recognize the mathematical meaning of common words and phrases, and relate everyday language to mathematical language and symbols.
Communicate mathematical thinking by using everyday language and appropriate mathematical language

By the end of the 3 - 4 program:

Apply and justify the use of a variety of problem solving strategies; e.g., make an organized list, guess and check
Use an organized approach and appropriate strategies to solve multi-step problems.
Interpret results in the context of the problem being solved; e.g., the solution must be a whole number of buses when determining the number of buses necessary to transport students.
Use mathematical strategies to solve problems that relate to other curriculum areas and the real world; e.g., use a timeline to sequence events; use symmetry in artwork.
Link concepts to procedures and to symbolic notation; e.g., model 3 x 4 with a geometric array, represent one-third by dividing an object into three equal parts
Recognize relationships among different topics within mathematics; e.g., the length of an object can be represented by a number.
Use reasoning skills to determine and explain the reasonableness of a solution with respect to the problem situation.
Recognize basic valid and invalid arguments, and use examples and counter examples, models, number relationships and logic to support or refute.
Represent problem situations in a variety of forms (physical model, diagram, in words or symbols), and recognize when some ways of representing a problem may be more helpful than others
Read, interpret, discuss and write about mathematical ideas and concepts using both everyday and mathematical language.
Use mathematical language to explain and justify mathematical ideas, strategies and solutions.

By the end of the 5 - 7 program:

Clarify problem-solving situation and identify potential solution processes; e.g., consider different strategies and approaches to a problem, restate problem from various perspectives.
Apply and adapt problem-solving strategies to solve a variety of problems, including unfamiliar and non-routine problem situations.
Use more than one strategy to solve a problem, and recognize there are advantages associated with various methods.
Recognize whether an estimate or an exact solution is appropriate for a given problem situation.
Use deductive thinking to construct informal arguments to support reasoning and to justify solutions to problems.
Use inductive thinking to generalize a pattern of observations for particular cases, make conjectures, and provide supporting arguments for conjectures.
Relate mathematical ideas to one another and to other content areas; e.g., use area models for adding fractions, interpret graphs in reading, science and social studies.
Use representations to organize and communicate mathematical thinking and problem solutions.
Select, apply and translate among mathematical representations to solve problems; e.g., representing a number as a fraction, decimal or percent as appropriate for a problem.
Communicate mathematical thinking to other and analyze the mathematical thinking and strategies of others.
Recognize and use mathematical language and symbols when reading, writing and conversing with others.

By the end of the 8 - 10 program:

Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information, and set limits for acceptable solution.
Apply mathematical knowledge and skills routinely in other content areas and practical situations.
Recognize and use connections between equivalent representations and related procedures for a mathematical concept; e.g., zero of a function and the x-intercept of the graph of the function, apply proportional thinking when measuring, describing functions, and comparing probabilities.
Apply reasoning processes and skills to construct logical verifications or counter-examples to test conjectures and to justify and defend algorithms and solutions.
Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas.
Write clearly and coherently about mathematical thinking and ideas.
Locate and interpret mathematical information accurately, and communicate ideas, processes and solutions in a complete and easily understood manner.

By the end of the 11 - 12 program:

Construct algorithms for multi-step and non-routine problems.
Construct logical verifications or counter-examples to test conjectures and to justify or refute algorithms and solutions to problems.
Assess the adequacy and reliability of information available to solve a problem.
Select and use various types of reasoning and methods of proof.
Evaluate a mathematical argument and use reasoning and logic to judge its validity.
Present complete and convincing arguments and justifications, using inductive and deductive reasoning, adapted to be effective for various audiences.
Understand the difference between a statement that is verified by mathematical proof such as theorem, and one that is verified empirically using examples or data.
Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations.
Communicate mathematical ideas orally and in writing with a clear purpose and appropriate for a specific audience.
Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation.

Activities/Examples:

1.) An easy way to incorporate math and problem solving together is by including problem solving questions in each unit, and on quizzes and tests, these questions can include:

Shawn bought a car for $5,600. He sold it to Rachael for 5/6 the price he paid for it. Rachael sold it to Ray for 1/5 less than she paid for it. Ray sold it to Rick for 3/4 what he paid. What did Rick pay for the car?
Everyday when Lisa returns from school she puts her change from buying lunch into a jar on her dresser. This weekend she decided to count her savings. She found that she had 72 coins—all nickels and dimes. The total amount was $4.95. How many coins of each kind did she have? See if you can find 2 different ways to solve this. Don't forget to explain your process.
From 11 positive integer scores on a 10-point quiz, the mean is 8, the median is 8, and the mode is 7. Find the maximum number of perfect scores possible on this test.

2.) Another not only easy but great way to engage students in problem solving is to use it when encountering everyday situations. Here are some examples:

The class is having a pizza party, and there are 24 people in the class (including the teacher). Everyone is to get 2 slices of pizza, but everyone has different preferences for toppings. Of the 24 people, 8 of them want pepperoni on their pizza, 12 people want cheese, and 4 want sausage. If each pizza contains 12 slices, how many pizzas must be ordered? And how should the toppings be distributed on the pizzas? In other words, what fraction of the total pizza orderwill contain pepperoni? Cheese? And sausage?
- This problem could even be taken a step further to include: In terms of pizzas, how many pizzas will contain pepperoni? Cheese? And sausage? (Keep in mind that not all the numbers will be whole numbers. Some will be fractions and/or mixed numbers.)

If I have 47 cents in my pocket consisting of 5 coins, what are the coins?
- This example could even be taken a step further by rewording the question to say, "If I have 47 cents in my pocket, what are the possible number of coins that I could have in my pocket, and what would they be?" In working out the problem, students would realize that there is more than one way to arrive at a total of 47 cents.

A recipe that you want to make for oatmeal raisin cookies calls for 1/2 cup of sugar, 2 sticks (8 ounces) of butter, 2 eggs, 1 teaspoon vanilla, 1 and 1/2 cups of flour, 1 teaspoon cinnamon, 1 teaspoon baking soda, 3 cups of oatmeal (uncooked), and 1 cup of raisins. However, you suddenly realize that you only have 3/4 cup of flour left. Can you cut down the recipe and still maintain the proper ratio of ingredients? Why or why not? If you can, then what fraction of the recipe would you make, and how much of each ingredient would you need to use?
- This problem could then be altered to ask: What if instead of 3/4 cup of flour, you had 1 cup of flour left? Could you use exactly one cup of flour and still maintain the proper ratio of ingredients, keeping in mind that the measuring tools you have available are: 1 cup, 1/3 cup, 1/2 cup, 1/4 cup, 1 teaspoon, 1/2 teaspoon, 1/4, teaspoon, and 1/8 teaspoon? Why or why not?